Method and apparatus for resource allocation and scheduling

ABSTRACT

A method and apparatus for resource allocation and scheduling within a wireless communication system is provided herein. During resource allocation, a maximum total system transmit power (P), a maximum number of codes available (N), a maximum number of codes for each user in the system (N=(N 1 , . . . , N d )), a maximum SINR value (S=(S 1 , . . . , S d )) for each user in the system, and a SINR per watt of transmit power for each user in the system (e=(e 1 , . . . , e d )) is received by a scheduler. Scheduler then outputs an optimal number of codes per user (n) and power levels per user to).

REFERENCE(S) TO RELATED APPLICATION(S)

The present application claims priority from provisional application,Ser. No. 60/523,971, entitled “METHOD AND APPARATUS FOR RESOURCEALLOCATION AND SCHEDULING,” filed Nov. 21, 2003, which is commonly ownedand incorporated herein by reference in its entirety.

FIELD OF THE INVENTION

The present invention relates generally to resource allocation andscheduling and in particular, to resource allocation and scheduling inan over-the-air communication system.

BACKGROUND OF THE INVENTION

Resource allocation and scheduling within communication systems isgenerally known in the art. During resource allocation and scheduling, asystem controller determines channel parameters for each user, alongwith a schedule for data transmission. For example, each user within acommunication system may be assigned a coding scheme, a power, a numberof codes utilized, and a data rate. As known, a communication systemgenerally comprises many users each having their own transmissionparameters. A challenge becomes optimizing the communication system byappropriately scheduling each user, and picking appropriate systemparameters for each user. Therefore, a need exists for a method andapparatus for scheduling and resource allocation within a communicationsystem that generally optimizes system performance.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a transmission circuitry in accordance withan embodiment of the present invention.

FIG. 2 is a flow chart showing operation of the circuitry of FIG. 1 withan embodiment of the present invention.

FIG. 3 is a flow chart showing operation of the circuitry of FIG. 1 withanother embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

To address the need for resource allocation and scheduling, a method andapparatus for resource allocation and scheduling within a wirelesscommunication system is provided herein. During resource allocation, amaximum total system transmit power (P), a maximum number of codesavailable (N), a maximum number of codes for each user in the system(N=(N₁, . . . , N_(d))), a maximum SINR value (S=(S₁, . . . , S_(d)))for each user in the system, and a SINR per watt of transmit power foreach user in the system (e=(e₁, . . . , e_(d))) is received by ascheduler. Scheduler then outputs an optimal number of codes per user(n) and power levels per user (p) based on P, N, N, and S.

The present invention encompasses a method for resource allocation. Themethod comprises the steps of receiving a maximum total system transmitpower (P), receiving a maximum number of system codes available (N),receiving a maximum number of codes for each user in the system (N),receiving a maximum SINR value (S) for each user in the system,receiving a SINR per watt of transmit power value for each user in thesystem (e), determining an optimal number of codes per user (n) andoptimal power levels per user (p) based on P, N, N, S, and e, andscheduling users with operating parameters based on n and p.

The present invention additionally encompasses a method for resourceallocation. The method comprises the steps of determining a value for anoptimal number of codes per user (n) within a communication system, anddetermining an optimal rate per user (r) and an optimal power per userto) based on n.

The present invention additionally encompasses an apparatus comprising ascheduler receiving a maximum total system transmit power (P), a maximumnumber of system codes available (N), a maximum number of codes for eachuser in the system (N), a maximum SINR value (S) for each user in thesystem, and a SINR per watt of transmit power value for each user in thesystem (e), the scheduler outputting an optimal number of codes per user(n) and optimal power levels per user to) based on P, N, N, S, and e.

Turning now to the drawings, wherein like numerals designate likecomponents, FIG. 1 is a block diagram of transmission circuitry 100. Asshown, transmission circuitry 100 comprises scheduler/optimizer 101,controller 103, and plurality of transmitters 105. During operation,system parameters enter scheduler 101. Scheduler 101 then computes anoptimized set of transmission parameters for each transmitter 105 andpasses this information to controller 103. During data transmission,controller 103 receives data destined to individual users, and utilizesthe set of optimized transmission parameters to appropriately controltransmitters 105. When data is passed to transmitters 105, transmitters105 utilize the transmission parameters to appropriately transmit datato individual users.

As an example, scheduler/optimizer 101 receives a set of users fromcontroller 103 and selects the users for transmission. For each of theusers selected, scheduler/optimizer 101 chooses a physical-layeroperating point (PLOP) which comprises a particular modulation andcoding scheme (MCS), a particular transmit power, a particular number ofcodes, and a particular information rate. Each PLOP consumes differentamounts of overall code and power resources.

Scheduler/optimizer 101 picks the “optimal” PLOP for each user such thatthe overall system is optimized. In particular, the optimal PLOP foreach user is chosen so that a sum of a weighted combination of the ratesassigned to users is maximized. The preferred value for the weightvector is the gradient of the utility function at the current throughputestimate

As discussed above, scheduler/optimizer 101 has the task of optimizingoverall system performance based on a given set of input parameters. Inthe preferred embodiment of the present invention this is accomplishedby optimizing a scheduling and resource allocation algorithm. Thefollowing text describes, 1) the generation of the resource allocationalgorithm, and 2) the optimization of the resource allocation algorithm.

1. Generation of the Resource Allocation Algorithm

Simply stated, optimizing a scheduling algorithm involves maximizing thedot product of the “gradient of utility” (as a function of thethroughput and the queue length) defined as ΔU(W_(t),Q_(t)):=φΔ_(w)U(W_(t),Q_(t))−Δ_(Q)U(W_(t),Q_(t)) and the rate vectorr_(t) to be selected at each scheduling time t, where:

U is a utility function,

-   -   W_(t)=(W_(1,t), . . . , W_(d,t)) is an estimate of user        throughput,    -   Q_(t)=(Q_(1,t), . . . , Q_(d,t)) is the user queue length or        delay,    -   Δ_(w)U(W₁, Q₁) is a vector of partial derivatives of U with        respect to W,    -   Δ_(Q)U(W₁, Q₁) is a vector of partial derivatives of U with        respect to Q,    -   r_(t)=(r_(1,t), . . . , r_(d,t)) is a rate vector selected, and    -   d is the number of users.

That is, we would like to pick the rate vector from the state dependentrate region R(e_(t)), which has the highest projection on the gradientof the utility. The rate region depends on the channel state. While thechannel state is not perfectly known we assume there is some estimate ofit available through a quality (e.g., CQI) feedback, etc. In particular,we assume that we know the predicted SINR per watt of transmitted powere_(i,t) for each user i and that e_(t)=(e_(1,t), . . . , e_(d,t))captures the channel state${\max_{r_{t} \in {\mathcal{R}{(e_{t})}}}{{\nabla{U\left( {W_{t},Q_{t}} \right)}^{T}} \cdot r_{t}}} = {\max_{r_{t} \in {\mathcal{R}{(e_{t})}}_{i}}{{{\overset{.}{U}}_{i}\left( {W_{i,t},Q_{i,t}} \right)}r_{i,t}}}$

For a certain class of utility functions we have $\begin{matrix}{{{\overset{.}{U}}_{i}\left( {W_{i},Q_{i}} \right)}:={{\phi\frac{\partial}{\partial W_{i}}{U_{i}\left( {W_{i},Q_{i}} \right)}} - {\frac{\partial}{\partial Q_{i}}{U_{i}\left( {W_{i},Q_{i}} \right)}}}} & (1) \\{\quad{= {{\phi\quad c_{i}W_{i}^{\alpha - 1}} + {b_{i}Q_{i}^{p - 1}}}}} & \quad \\{\quad{\text{=}\text{:}\quad w_{i}}} & \quad\end{matrix}$where

-   -   c_(i) is the QoS weight associated with the throughput for user        i,    -   d_(i) is the QoS weight associated with the queue length for        user i,    -   φ is filter coefficient associated with the throughput,    -   α is a fairness parameter associated with the throughput,    -   p is a fairness parameter associated with the queue length.        Thus the optimization problem results in        $\max_{r \in {\mathcal{R}{(e)}}}{\sum\limits_{i}{w_{i}r_{i}}}$        where w_(i) is given by equation (1) above and where we have        suppressed the dependence on time t for convenience. Note that        selection of the rate r_(i) is coupled with the selection of the        physical layer operating point (PLOP) that includes the number        of codes n_(i) and the power p_(i). These PLOP variables need to        satisfy system and per user constraints such as n_(i) ≤ N_(i)        ${\sum\limits_{i}n_{i}} \leq N$ ${\sum\limits_{i}p_{i}} \leq P$        and these are part of the description of the state dependent        rate region R(e). As described in U.S. Patent Publication No. US        2002/0147022 “Method for Packet Scheduling and Radio Resource        Allocation in a Wireless Communication System,” by Agrawal, et        al., the rate per code can be written in terms of the signal to        noise ratio (SINR) per code so that        $\frac{r_{i}}{n_{i}} = {\Gamma\left( {SINR}_{i} \right)}$        where the SINR per code is given by        ${SINR}_{i} = {\frac{p_{i}}{n_{i}}e_{i}}$        and where Γ is some appropriate function. A reasonable choice of        the function Γ is the well known information theoretic channel        capacity formula Γ(x)=Blog(1+x). Such a functional relation        between the rate and the SINR also provides a good curve fit to        the High-Speed Downlink Packet Access (HSDPA) MCS table with B        equal to the symbol rate (chip rate/spreading factor) which for        HSDPA is B=3840000/16=240000 symbols/sec. Note also that for the        best curve fit, the SINR is decreased by a factor of A=1.7,        i.e., Γ(x)Blog(1+x/A) is used. Plugging in this formula results        in the following equation for the rate r_(i) in terms of e_(i),        n_(i), p_(i): $\begin{matrix}        {r_{i} = {n_{i}B\quad{\log\left( {1 + \frac{p_{i}e_{i}}{{An}_{i}}} \right)}}} & (2)        \end{matrix}$        Note that the SINR per watt of transmit power on a HS-PDSCH code        is given by:        $e_{i} = {\left( \frac{E_{c}}{N_{t}} \right)_{pilot}\frac{16}{P_{pilot}}}$        For notational simplicity e_(i) is redefined to be e_(i)/A the        “effective” SINR per watt of transmit power on a HS-PDSCH code,        i.e.,        $e_{i} = {\left( \frac{E_{c}}{N_{t}} \right)_{pilot}\frac{16}{P_{pilot}}\frac{1}{A}}$        The rate region is then given by $\begin{matrix}        {{\mathcal{R}(e)} = \left\{ {{{r \geq {0\text{:}\quad r_{i}}} = {n_{i}B\quad{\log\left( {1 + \frac{p_{i}e_{i}}{n_{i}}} \right)}}},} \right.} & \quad \\        \left. \quad{{n_{i} \leq {N_{i}{\forall i}}},{{\sum\limits_{i}n_{i}} \leq N},{{\sum\limits_{i}p_{i}} \leq P}} \right\} & \quad        \end{matrix}$        By redefining w_(i) to be        $w_{i} = \frac{B\left( {{\phi\quad c_{i}W_{i}^{\alpha - 1}} + {b_{i}Q_{i}^{p - 1}}} \right)}{\ln\quad 2}$        where    -   w=(w₁, . . . , w_(d)) is the gradient vector,    -   W=(W₁, . . . , W_(d)) is estimated throughput vector for all the        users,    -   Q=(Q₁, . . . , Q_(d)) is estimated throughput vector for all the        users,    -   c_(i) is the QoS weight (possibly 0) associated with the        throughput for user i,    -   d_(i) is the QoS weight (possibly 0) associated with the queue        length for user i,    -   φ is filter coefficient associated with the throughput,    -   α is a fairness parameter associated with the throughput,    -   p is a fairness parameter associated with the queue length,    -   B is a scaling constant to match up the formula to bits.        This is a specific instance of the weight vector w=(w₁, . . . ,        w_(d)); however, in general w_(i) cab be any function of the        throughput W_(i) and queue length Q_(i) which is non-increasing        in the throughput and non-decreasing in the queue length. The        function may also take as parameters various QoS related        information such as delay requirement, QoS class, bandwidth        requirement, etc.

The optimization problem can be rewritten asV*:=max_((n,p)εχ) V(n,p)   (3)subject to $\begin{matrix}{{\sum\limits_{i}n_{i}} \leq N} & \quad \\{{\sum\limits_{i}p_{i}} \leq {P\quad{where}}} & \quad \\{{{V\left( {n,p} \right)}:={\sum\limits_{i}{w_{i}n_{i}{\ln\left( {1 + \frac{p_{i}e_{i}}{n_{i}}} \right)}}}},} & (4) \\{{\mathcal{X}:=\left\{ {\left( {n,p} \right) \geq {0\text{:}\quad n_{i}} \leq {N_{i}\quad{\forall i}}} \right\}},} & \quad\end{matrix}$

-   -   n is a vector of code allocations, i.e. a vector comprising        codes per user,    -   p is a vector of power allocations, i.e., a vector comprising        power levels per user,    -   w is the vector of w_(i)s, and    -   e is the vector of e_(i)s.        Note that the constraint set χ is convex. It can also be        verified that V is concave in (n,p). In case of a retransmission        for a user i the “effective” SINR per watt per code e_(i) of        that user may be bumped up by a suitable factor.

In addition to the constraints captured above, the following per userpower constraints are allowed, any of which may be rendered redundant bya suitable setting of the parameters.

-   -   peak power constraint:        p_(i)≦P_(i), ∀i.    -   SINR constraint:        ${{SINR}_{i} = \left. {\frac{p_{i}e_{i}}{n_{i}} \leq S_{i}}\Leftrightarrow{p_{i} \leq {S_{i}\frac{n_{i}}{e_{i}}}} \right.},{\forall{i.}}$    -   rate per code (note that the encoder rate assuming a fixed        modulation order is a constant times this, viz. the encoder rate        is the rate per code divided by the coded output rate which        itself is the product of modulation order (2 or 4) times the        symbol rate (240000 symbols/sec for HSDPA))    -   rate $\begin{matrix}        {\frac{r_{i}}{n_{i}} = \left. {{\ln\left( {1 + \frac{p_{i}e_{i}}{n_{i}}} \right)} \leq \left( {R/N} \right)_{i}}\Leftrightarrow{p_{i} \leq {\left( {{\mathbb{e}}^{{({R/N})}_{i}} - 1} \right)\frac{n_{i}}{e_{i}}\quad{\forall{i.}}}} \right.} & (5) \\        {r_{i} = \left. {{n_{i}{\ln\left( {1 + \frac{p_{i}e_{i}}{n_{i}}} \right)}} \leq R_{i}}\Leftrightarrow{p_{i} \leq {\left( {{\mathbb{e}}^{R_{i}/n_{i}} - 1} \right)\frac{n_{i}}{e_{i}}\quad{\forall{i.}}}} \right.} & \quad        \end{matrix}$

All of the above per user power constraints are special cases of$\begin{matrix}{{{SINR}_{i} = {\frac{p_{i}e_{i}}{n_{i}} \leq {s_{i}\left( n_{i} \right)}}},{\forall i},} & (6)\end{matrix}$where the function s_(i) is also dependent on the fixed (for theoptimization problem) parameters P_(i),e_(i),S_(i),R_(i),(R/N)_(i). Infact all of the above constraints can be combined into the singleconstraint (6) above with${s_{i}\left( n_{i} \right)} = {\min\left\{ {\frac{P_{i}e_{i}}{n_{i}},S_{i},\left( {{\mathbb{e}}^{{({R/N})}_{i}} - 1} \right),\left( {{\mathbb{e}}^{R_{i}/n_{i}} - 1} \right)} \right\}}$Two special cases of this of interest are:

-   -   1. s_(i)(n_(i))≡s_(i) does not depend on n_(i).    -   2. s_(i)(n_(i))≡s_(i)=∞.

With the per user power constraints, the constraint set χ is furtherrestricted toχ:={(n,p)≧0: n _(i) ≦N _(i) , p _(i) ≦s _(i)(n _(i))n _(i) /e _(i)∀i}  (7)

The constraint set continues to be convex if s_(i)(n_(i))n_(i) is aconcave function of n_(i). Note that s_(i)(n_(i))n_(i) is indeed aconcave function of n_(i) for the two special case (1-2) mentionedabove. However, for the rate constraint case (equation 18),s_(i)(n_(i))n_(i) is convex in n_(i).

1.1 Handling Retransmissions

As mentioned earlier, users with retransmissions need to be treateddifferently in two respects:

-   -   1. T heir effective SINR per watt per code needs to be        increased.    -   2. T hey have an equality constraint on their rate r_(i)=R_(i)

The first can be handled easily by adjusting their e_(i) appropriately.To handle the latter constraint, we observe that the rate constraint isequivalent to $\begin{matrix}{r_{i} = {{\frac{B}{\ln\quad 2}n_{i}{\ln\left( {1 + \frac{p_{i}e_{i}}{n_{i}}} \right)}} = {\left. R_{i}\Leftrightarrow p_{i} \right. = {\frac{n_{i}}{e_{i}}\left( {{\mathbb{e}}^{\frac{R_{i}\ln\quad 2}{{Bn}_{i}}} - 1} \right)}}}} & (8)\end{matrix}$

2. Operation of Scheduler/Optimizer

Scheduler/optimizer 101 receives a set of parameters and solves for V*by optimizing the resource allocation algorithm of equation (3). Moreparticularly, scheduler/optimizer 101 receives

-   -   P=Maximum total system transmit power,    -   N=Maximum total codes available to the system,    -   N=Vector comprising the maximum number of codes (e.g., spreading        codes) for each user in the system,    -   R=Vector comprising the maximum data rate for each user in the        system,    -   R/N=Vector comprising the maximum data rate per code for each        user in the system,    -   P=Vector comprising the maximum transmit power for each user in        the system,    -   S=Vector comprising the maximum noise (e.g., SINR) for each user        in the system,    -   e=Vector comprising the noise (e.g., SINR) per power (e.g., SINR        per watt) for each user in the system,    -   W=Vector comprising the estimated throughput vector for each        user in the system,    -   Q=Vector comprising the queue length vector for each user in the        system,    -   c=Vector comprising the QoS weight associated with the        throughput for each user in the system, and    -   b=Vector comprising the QoS weight associated with the queue        length for each user in the system.

In the preferred embodiment of the present invention scheduler/optimizer101 determines optimal values of r_(i), n_(i), and p_(i) for each useri, outputting vectors r, n, and p for communication system 100. Asdiscussed above, vectors r, n, and p for communication system 100 aredetermined by solving for V* in equation (3) given the set of inputparameters described above. Scheduler/optimizer 101 solves for V*subject to the additional constraints described above in (7).

2.1 FIRST EMBODIMENT

In case ${{\sum\limits_{i}N_{i}} \leq N},{n_{i} = N_{i}}$is the optimal code allocation, we are left with just a poweroptimization problem which can be solved very easily as shown later.Henceforth, we will tackle the case when this is not the case, i.e.,${\sum\limits_{i}N_{i}} > {N.}$Also, we allow n_(i) to take on non-integer values as a relaxation tothe integer optimization problem. We find that in most cases, theoptimal solution turns out to assign integer values to n_(i). We solvethe optimization problem by looking at the dual formulation. Define theLagrangian${L\left( {p,n,\lambda,\mu} \right)} = {{\sum\limits_{i}{w_{i}n_{i}{\ln\left( {1 + \frac{p_{i}e_{i}}{n_{i}}} \right)}}} + {\lambda\left( {P - {\sum\limits_{i}p_{i}}} \right)} + {\mu\left( {N - {\sum\limits_{i}n_{i}}} \right)}}$Based on this we can define the dual functionL(λ,μ):=max_((n,p)εχ) L(p,n,λ,μ)   (9)where

-   -   χ is set of valid choices for (n,p) given by the constraint set        (7),    -   λ is Lagrange multiplier for the power constraint, and    -   μ is Lagrange multiplier for the code constraint.        The dual problem is to find        L*:=min_((λ,μ)≧0) L(λ,μ)   (10)        Also, define        L(λ):=min_(μ≧0) L(λ,μ)=max_((n,p)εχ) L(p,n,λ,μ)   (11)

For this part of the solution we will ignore the rate constraint (5)((8) for retransmissions) or replace it with a linear constraint asdescribed below in “Handling rate restrictions.”

Note: Handling Rate Restrictions

The constraint set with the rate restriction (5) or (8) is non-convex,which would not allow us to optimize over n_(i) very easily. To simplymatters, we may assume that n_(i)=N_(i) in the second term on the right.Thus we assume that (5) is replaced by$p_{i} \leq {\frac{n_{i}}{e_{i}}\left( {e^{\frac{R_{i}\ln\quad 2}{{BN}_{i}}} - 1} \right)}$In case of constraint (8), we can replace it by$p_{i} = {\frac{n_{i}}{e_{i}}{\left( {e^{\frac{R_{i}\ln\quad 2}{{BN}_{i}}} - 1} \right).}}$

This may be justified on a few different grounds. First, we haveobserved that in many cases the optimal n_(i)=N_(i). Secondlysubstituting n_(i)=N_(i) makes things a bit worse for this user, but thebump up in e_(i) (for the case of retransmissions) will increase thew_(i)e_(i) of this user which should probably still help select thisuser. We could also try putting a different linear approximation to theabove constraint. Note that this constraint is only needed for the codeallocation part. Some other heuristics are also possible. For instancewe may replace the rate constraint as above and if we still pick thisuser and it gets n_(i)=N_(i), then we have most likely not sacrificedanything. Similarly we could try n_(i)=1 in the second term which shouldmake the user more likely to be selected. If it is still not selected,once again we have probably not sacrificed anything by thisapproximation. However for the cases in between (which would probably berare), we may want to explicitly vary the n_(i) for this user explicitlyand optimize. Note that once the code allocation is made, we can workwith the original rate constraint for the optimizations over powers.

Since the objective function is concave and the constraint set isconvex, there is no duality gap allowing us to use the solution of thedual to compute the solution of the primal.

Based on this the scheduler/optimizer 101 determines appropriate valuesfor r, n, and p via the following steps (also illustrated in FIG. 2):

-   -   1. Collect all the input parameters P, N, N, R, R/N, P, S, e, W,        Q, c, b. (Step 201)    -   2. For a fixed λ and μ find the optimal (n,p) analytically.        (Step 203)    -   3. Then keeping λ fixed, find the optimal μ*(μ). (Step 205)    -   4. Also, keeping λ fixed, calculate L(λ) as defined in equation        (11). (Step 207).    -   5. Numerically search for the optimal λ=λ*, in other words,        search for the value of λ that minimizes L(λ) to obtain L*.        (Step 209)    -   6. Calculate n=n*(λ*,μ*(λ*)) (Step 211)    -   7. Calculate optimal p and r keeping n fixed. (Step 213)

FIG. 2 is a flow chart showing operation of scheduler/optimizer 101. Thelogic flow begins at step 201 where e, P, N, N, and S are collected bythe optimizer 101. At step 203 n and p are calculated analytically bykeeping λ and μ fixed. In particular λ and μ are fixed but arbitrary andn and p are calculated as a function of λ and μ to optimize theLagrangian in order to obtain the dual function as shown in equation(9).

At step 205 λ is fixed but arbitrary and an optimal μ*(λ) is determinedso that we minimize the dual function L(λ,μ) to obtain L(λ). L(λ) iscalculated per equation 23 at step 207 and a numerical search for anoptimal λ=λ* occurs at step 209. In other words, at step 209 λ=λ* isdetermined so that L*=L(λ*)=min_(λ≧0)L(λ). At step 211 n=n*(λ*,μ*(λ*))is calculated per equation (14). The vector R is collected and r and pare calculated keeping n fixed (step 213). In particular n is fixed tothe value obtained in step 5, and p and r are calculated as perequations (21) and (2) taking the actual rate constraint R into account.Finally, at step 215 values for n, p and r are output to controller 103and utilized to schedule and transmit data via transmitters 105.

The following discussion further details the steps taken above in FIG.2.

2.1.1 Step 203, for Fixed λ and μ Find the Optimal (n,p) Analytically

Optimizing the Lagrangian (in the dual) for a fixed λ, μ, and n (thisincludes the optimization over powers assuming a fixed feasible n) weget $p_{i}^{*} = {\frac{n_{i}}{e_{i}}\left\{ \begin{matrix}{0,} & {{w_{i}e_{i}} \leq \lambda} \\{{\frac{w_{i}e_{i}}{\lambda} - 1},} & {\frac{w_{i}e_{i}}{1 + {s_{i}\left( n_{i} \right)}} \leq \lambda < {w_{i}e_{i}}} \\{{s_{i}\left( n_{i} \right)},} & {\lambda < \frac{w_{i}e_{i}}{1 + {s_{i}\left( n_{i} \right)}}}\end{matrix} \right.}$or equivalently, $\begin{matrix}{p_{i}^{*} = {{\frac{n_{i}}{e_{i}}\left\lbrack {\min\left\{ {{\frac{w_{i}e_{i}}{\lambda} - 1},{s_{i}\left( n_{i} \right)}} \right\}} \right\rbrack}^{+}.}} & (12)\end{matrix}$The resulting SINR per code is given by $\begin{matrix}{\frac{p_{i}^{*}e_{i}}{n_{i}} = {{s^{*}\left( {\frac{w_{i}e_{i}}{\lambda},{s_{i}\left( n_{i} \right)}} \right)}:=\left\lbrack {\min\left\{ {{\frac{w_{i}e_{i}}{\lambda} - 1},{s_{i}\left( n_{i} \right)}} \right\}} \right\rbrack^{+}}} & (13)\end{matrix}$

The last case is vacuous when we have no per user constraints on thepowers, i.e., s_(i)(n_(i))=∞. Note that in case s_(i)(n_(i))≡s_(i) doesnot depend on n_(i), the resulting SINR per code also does not depend onthe number of codes n_(i) and the power requirement scales linearly inthe number of codes.

Define${h\left( {{w_{i}e_{i}},{s_{i}\left( n_{i} \right)},\lambda} \right)}:=\left\{ \begin{matrix}{0,} & {{w_{i}e_{i}} \leq \lambda} \\{\left\lbrack {\frac{\lambda}{w_{i}e_{i}} - 1 - {\ln\quad\frac{\lambda}{w_{i}e_{i}}}} \right\rbrack,} & {\frac{w_{i}e_{i}}{1 + {s_{i}\left( n_{i} \right)}} \leq \lambda < {w_{i}e_{i}}} \\{\left\lbrack {{\ln\left( {1 + {s_{i}\left( n_{i} \right)}} \right)} - {\frac{\lambda}{w_{i}e_{i}}{s_{i}\left( n_{i} \right)}}} \right\rbrack,} & {\lambda < \frac{w_{i}e_{i}}{1 + {s_{i}\left( n_{i} \right)}}}\end{matrix} \right.$Then for case (1) s_(i)(n_(i))≡s_(i) does not depend on n_(i).Additionally,h(w _(i) e _(i) , s _(i) (n _(i)),λ)=h(w _(i) e _(i) , s _(i), λ)also does not depend on n_(i). For this case it is easy to furtheroptimize the Lagrangian over n_(i) to calculate the dual function. Inparticular, we find that $\begin{matrix}{n_{i}^{*} = {{n_{i}^{*}\left( {\lambda,\mu} \right)} = \left\{ \begin{matrix}{0,} & {{w_{i}{h\left( {{w_{i}e_{i}},s_{i},\lambda} \right)}} < \mu} \\{{0 \leq n_{i} \leq N_{i}},} & {{w_{i}{h\left( {{w_{i}e_{i}},s_{i},\lambda} \right)}} = \mu} \\{N_{i},} & {{w_{i}{h\left( {{w_{i}e_{i}},s_{i},\lambda} \right)}} > \mu}\end{matrix} \right.}} & (14)\end{matrix}$Note that for the case μ=w_(i)h(w_(i)e_(i), s_(i), λ) any 0≦n_(i)N_(i)is optimal. Then the dual function is given by $\begin{matrix}{{L\left( {\lambda,\mu} \right)} = {{\sum\limits_{i}{\left\lbrack {{\mu_{i}(\lambda)} - \mu} \right\rbrack^{+}N_{i}}} + {\mu\quad N} + {\lambda\quad P}}} & (15)\end{matrix}$whereμ_(i)(λ):=w _(i) h(w _(i) e _(i) ,s _(i),λ)

2.1.2 Step 205, Fixing λ and Optimizing μ.

In order to optimize the dual function (15) over μ, we sort the users indecreasing order of μ_(i)(λ). Let j* be the smallest integer such that$\begin{matrix}\begin{matrix}{{\sum\limits_{i = 1}^{j^{*}}N_{i}} \geq N} \\{{L(\lambda)}:={\min_{\mu \geq 0}{L\left( {\lambda,\mu} \right)}}} \\{\quad{= {{\sum\limits_{i = 1}^{j^{*} - 1}{{\mu_{i}(\lambda)}N_{i}}} + {{\mu_{j^{*}}(\lambda)}\left( {N - {\sum\limits_{i = 1}^{j^{*} - 1}N_{i}}} \right)} + {\lambda\quad P}}}}\end{matrix} & (16)\end{matrix}$and the minimizing μ is given by $\begin{matrix}{{\mu^{*}(\lambda)}:={\mu_{j^{*}}(\lambda)}} & (17)\end{matrix}$

Note that μ_(j)(λ)≧μ_(j+1)(λ) by the above ordering. Thus μ*(λ) is athreshold; users with their μ_(i)(λ)>μ*(λ), get their full codecapability and those with μ_(i)(λ)<μ*(λ) get none. Also note that whenw_(i)≧w_(j) and e_(i)>e_(j), μ_(i)(λ)≧μ_(j)(λ), and user i will be givena full code allocation before allocating any codes to user j. Thisimplies that when the fairness parameter α=1 (the “max C/I scheduler”)and all users have the same QoS weight, w_(i)'s are constant andidentical across users and thus packing users into the code budget inorder of their e_(i)'s is optimal. This is illustrated below withrespect to description of the alternate embodiment of the presentinvention.

In case μ_(j−1)*(λ)>μ_(j)*(λ)>μ_(j+1)*(λ), there is a unique feasible n*that satisfies the sum code constraint with equality and that optimizesthe Lagrangian for μ=μ*(λ). It is given by $\begin{matrix}{n_{i}^{*} = \left\{ {\begin{matrix}{N_{i},} & {i < j^{*}} \\{{N - {\sum\limits_{i = 1}^{j^{*} - 1}N_{i}}},} & {i = j^{*}} \\{0,} & {i > j^{*}}\end{matrix}.} \right.} & (18)\end{matrix}$However, if this is not the case, there is a tie and there are multiplen* that optimize the Lagrangian. However, not all of these choices of n*and the resulting p* given by (18) and (12) will be feasible. In orderto achieve feasibility we must also satisfy the additional code andpower constraints. Let n*(λ) denote any of the maximizing n constructedabove.

We have just shown that the optimal code allocation has the followingproperties:

-   -   1. For the case of N_(i)=N at most two users will be scheduled.    -   2. If all N_(i) are equal, then at most ┌N/N_(i)┐+1 users will        be scheduled. All but two users will have their full code        allocation.    -   3. In general all but two users will have their full code        allocation.

2.1.3 Step 213 Calculating r and p While Keeping n Fixed

In step 213 we solve for the optimal powers given a code allocation n.It should be noted that n may be derived as discussed above withreference to steps 203 and 205, or alternatively, n may bepre-determined and provided as an input into the power-optimizationalgorithm. Denote byV*(n):=max_({p≧0: p) _(i) _(≦s) _(i) _((n) _(i) _()n) _(i) _(/e) _(i)_(∀i}) V(n,p)subject to ${\sum\limits_{i}p_{i}} \leq {P.}$

This is solved by finding λ*(n) using the dual formulation and thencomputing the optimal p*(n) as described before. Since n is fixed thereare no restrictions on the function n_(i)s_(i)(n_(i)) except that it hasto be non-negative. Thus, we can incorporate the rate constraints in thecalculations performed in this step.

Without loss of generality we remove any users with zero codeallocations. Let M be the number of users with positive code allocation.We first need to check if the sum power constraint is inactive, i.e.,${\sum\limits_{i}p_{i}} = {{\sum\limits_{i}{\frac{n_{i}}{e_{i}}{s_{i}\left( n_{i} \right)}}} \leq {P.}}$If this is the case the optimal power allocations are just theindividual power constraints$p_{i}^{*} = {\frac{n_{i}}{e_{i}}{s_{i}\left( n_{i} \right)}}$and we are done.

Henceforth, we proceed when this constraint is active. In this case wecan show that the sum power constraint must be satisfied with equalityfor the optimal powers (otherwise at least one of the users' powers canbe increased resulting in a higher primal value function). We can nowconstruct the Lagrangian in the sum power constraint alone.${L_{p}\left( {p,\lambda} \right)}:={{\sum\limits_{i}{w_{i}n_{i}\quad{\ln\left( {1 + \frac{p_{i}e_{i}}{n_{i}}} \right)}}} + {\lambda\left( {P - {\sum p_{i}}} \right)}}$In case ${{\sum\limits_{i}n_{i}} = N},$the two Lagrangians are equal. The dual function is given byL _(p)(λ):=max_({p≧0: p) _(i) _(≦s) _(i) (n _(i) _()n) _(i) _(/e) _(i)_(∀i}) L _(p)(p,λ).

Note that for optimizing over powers, the constraint set is alwaysconvex regardless of the function s_(i)(n_(i))n_(i). MaximizingL_(p)(p,λ) over p is essentially the same as the problem for L(p,n,λ,μ)as covered in the previous section. The optimal p is given by (12) asbefore, i.e., $\begin{matrix}{{p_{i}^{*}\left( {n,\lambda} \right)} = {{\frac{n_{i}}{e_{i}}\left\lbrack {\min\left\{ {{\frac{w_{i}e_{i}}{\lambda} - 1},{s_{i}\left( n_{i} \right)}} \right\}} \right\rbrack}^{+}.}} & (19)\end{matrix}$

In Agrawal et al. it was shown that λ*(n) optimizes L_(p)(λ) if and onlyif the corresponding p*(n,λ) satisfies${\sum\limits_{i}p_{i}^{*}} = {P.}$Substituting from (19) we get that λ*(n) is given by any solution of theequation: $\begin{matrix}{\lambda = {\frac{\sum\limits_{i}{n_{i}w_{i}1_{\{{\frac{w_{i}e_{i}}{1 + {s_{i}{(n_{i})}}} \leq \lambda < {w_{i}e_{i}}}\}}}}{P - {\sum\limits_{i}{\frac{n_{i}{s_{i}\left( n_{i} \right)}}{e_{i}}1_{\{{\lambda < \frac{w_{i}e_{i}}{1 + {s_{i}{(n_{i})}}}}\}}}} + {\sum\limits_{i}{\frac{n_{i}}{e_{i}}1_{\{{\frac{w_{i}e_{i}}{1 + {s_{i}{(n_{i})}}} \leq \lambda < {w_{i}e_{i}}}\}}}}}.}} & (20)\end{matrix}$

Then the optimal power allocation vector p*(n) is given by$\begin{matrix}{{p_{i}^{*}(n)} = {{p_{i}^{*}\left( {n,{\lambda^{*}(n)}} \right)} = {{\frac{n_{i}}{e_{i}}\left\lbrack {\min\left\{ {{\frac{w_{i}e_{i}}{\lambda^{*}(n)} - 1},{s_{i}\left( n_{i} \right)}} \right\}} \right\rbrack}^{+}.}}} & (21)\end{matrix}$Since λ occurs on both sides of equation (20) solving for λ*(n) may seemhard. We provide an algorithm solution to this as follows:

For users i=1, . . . ,M, define $\begin{matrix}{b_{i} = {w_{i}e_{i}}} \\{a_{i} = {\frac{w_{i}e_{i}}{1 + {s_{i}\left( n_{i} \right)}}.}}\end{matrix}$Note that a_(i)<b_(i). Sort the set {a_(i),b_(i); i=1, . . . ,2M} into adecreasing set of numbers {x[l];l=1, . . . ,2M}. For, each l=1, . . .,2Malso let y[l]:=i if x[l]=b_(i) or x[l]=a_(i), and let${z\lbrack l\rbrack}:=\left\{ {\begin{matrix}{1,} & {{{if}\quad{x\lbrack l\rbrack}} = a_{y{\lbrack l\rbrack}}} \\{0,} & {otherwise}\end{matrix}.} \right.$Ties are resolved arbitrarily. Next find the largest λ ε{x[l]; l=1, . .. ,2M} such that${{\sum\limits_{i}{p_{i}^{*}\left( {n,\lambda} \right)}} \geq P},$i.e., find the smallest l such that${\sum\limits_{i}{p_{i}^{*}\left( {n,{x\lbrack l\rbrack}} \right)}} \geq {P.}$In the paper “Joint Scheduling and Resource Allocation in CDMA Systems,”by Agrawal et al., we have shown that λ*(n) ε [x[l],x[l-1]) and is givenby the solution of equation (20). Moreover for λ in the above range, theindicators in equation (20) are not dependent on the particular value ofλ and hence we have an easily computable expression for λ.

Below we provide pseudo-code for an algorithm that finds the solution tothe above equation (20). The algorithm complexity is O(MlogM) due to asort.

Pseudo-Code for Preferred Embodiment of Solution λ*(n)

lambda () { Psum = 0; Ppossum = 0; Pmaxsum = 0; nwpossum = 0; for (i=1,i <= M, i++) { pos[i] = 0; max[i] = 0; } l = 1; while (Psum < P && l <2M) { i = y[l] if (z[l] = 1) { $\begin{matrix}{{{\max\lbrack i\rbrack} = 1};} \\{{{Pmaxsum}+={\frac{n_{i}}{e_{i}}{s_{i}\left( n_{i} \right)}}};} \\{{//{Pmaxsum}} = {\sum\limits_{i}\left( {\frac{n_{i}}{e_{i}}*{s_{i}\left( n_{i} \right)}*{\max\lbrack i\rbrack}} \right)}} \\{{{Ppossum}-=\frac{n_{i}}{e_{i}}};} \\{{//{Ppossum}} = {\sum\limits_{i}\left( {\frac{n_{i}}{e_{i}}*\left( {{{pos}\lbrack i\rbrack} - {\max\lbrack i\rbrack}} \right)} \right)}} \\{{{nwpossum}-={n_{i}*w_{i}}};} \\{{//{nwpossum}} = {\sum\limits_{i}\left( {n_{i}*w_{i}*\left( {{{pos}\lbrack i\rbrack} - {\max\lbrack i\rbrack}} \right)} \right)}}\end{matrix}\quad$ } else{ $\begin{matrix}{{{{pos}\lbrack i\rbrack} = 1};} \\{{{Ppossum}+=\frac{n_{i}}{e_{i}}};} \\{{//{Ppossum}} = {\sum\limits_{i}\left( {\frac{n_{i}}{e_{i}}*\left( {{{pos}\lbrack i\rbrack} - {\max\lbrack i\rbrack}} \right)} \right)}} \\{{{nwpossum}+={n_{i}*w_{i}}};} \\{{//{nwpossum}} = {\sum\limits_{i}\left( {n_{i}*w_{i}*\left( {{{pos}\lbrack i\rbrack} - {\max\lbrack i\rbrack}} \right)} \right)}} \\\}\end{matrix}\quad$ $\begin{matrix}{{l++};} \\{{{Psum} = {\sum\limits_{i}\left( {\frac{n_{i}}{e_{i}}s*\left( {\frac{w_{i}\quad e_{i}}{x\lbrack l\rbrack},{s_{i}\left( n_{i} \right)}} \right)} \right)}};} \\\}\end{matrix}\quad$ lambda = Ppossum/(P − Pmaxsum); it (lambda > x[l−1]or lambda > x[l]) { error! } else{ return (lambda) } }${where}\quad s*\left( {\frac{w_{i}e_{i}}{\lambda},{s_{i}\left( n_{i} \right)}} \right)\quad{is}\quad{defined}\quad{in}\quad{equation}\quad{(13).}$

2.1.3.1 Per User Equality Constraints

If for some user i, the inequality p_(i)≦s_(i)(n_(i))n_(i) is replacedby the equality constraint p_(i)=e_(i)s_(i)(n_(i))n_(i), as may happenfor instance in a retransmission when the rate is fixed, we may simplyremove this user from the problem after subtracting p_(i) from P andproceed as before.

2.1.3.2 No Per User Power Constraints

For the case that s_(i)(n_(i))=∞, the following simpler algorithm findsλ*(n). First, without loss of generality, assume that the users arenumbered in descreasing order of w_(i)e_(i). Calculate the following$\begin{matrix}{{\overset{\sim}{\lambda}}_{j} = \frac{\sum\limits_{i = 1}^{j}{w_{i}n_{i}}}{P + {\sum\limits_{i = 1}^{j}\frac{n_{i}}{e_{i}}}}} & {\forall{j \in \left\{ {1,2,\ldots\quad,M} \right\}}}\end{matrix}$We determine λ*(n) as follows:

-   -   1. Set j=1.    -   2. While w_(j+1)e_(j+1)>{tilde over (λ)}_(j), j++    -   3. λ*={tilde over (λ)}_(j)

2.1.4 Step 209, Obtaining L*

Any algorithm that minimizes or attempts to minimize L(λ) over λ isconsidered here. In the preferred embodiment we do so iteratively, wherein each step k of the iteration we narrow the range [λ_(k) ^(LB),λ_(k)^(UB)] of the optimal λ* in a manner such that starting at k=0 with 0≦λ₀^(LB)≦λ₀ ^(UB)<∞ and for some d≧1 and 0≦ρ<1,[λ_((k+1)d) ^(UB)−λ_((k+1)d) ^(LB)]≦ρ[λ_(kd) ^(UB)−λ_(kd) ^(UB) ] ∀k≧0.

A plurality of such schemes may be designed all of which guaranteegeometrically fast convergence to the optimal λ*. Two such embodimentsare described below.

2.1.4.1 First Embodiment for Finding λ*

The algorithm makes use of the functions L(λ) (11), n*(λ) (18) and thetie resolution, λ*(n) (20), p*(n) (21), and V(p,n) (4) defined earlier.The function r(p,n) is simply the rate given the powers and codes givenin (2). Pseudo-code for calculating λ*(n) is also given in Step 213.

In “Joint scheduling and resource allocation in CDMA systems,” by R.Agrawal, V. Subramanian, R. Berry, G. Casheekar, A. Diwan, B. Love, itwas shown that L(λ) is convex and that the optimalλ*=argmin_(λ≧0)L(λ)ε[λ^(min),λ^(max)] where λ^(min)=0 andλ^(max)=max_(i)w_(i)e_(i).

At the k^(th) iteration we identify λ* in the range [λ_(k) ^(LB),λ_(k)^(UB)] with the estimate of λ* given by λ_(k)*. The state of thealgorithm at any time k is given byS _(k)={[λ_(k) ^(LB) ,L _(k) ^(LB)], [λ_(k) *,L _(k) *,n _(k*)], [λ_(k)^(UB) ,L _(k) ^(UB)]}with the assumption that L_(k)*=L(λ_(k)*)≦min (L_(k) ^(UB)=L(λ_(k)^(UB)),L_(k) ^(LB)=L(λ_(k) ^(LB))).

Step 0 initializes the state of this algorithm and provides an exitcheck in case we got lucky to hit the optimal allocation.

Step 1 does the iteration on the above state of the algorithm and alsouses an exit check.

Step 2 polishes up by taking additional constraints into account andreoptimizing the powers, calculating rates/TBS etc.

The algorithm works as follows.

Step 0: Initialize

-   -   (a) λ₀ ^(LB)=0    -   (b) If there are no per user power constraints on at least one        user then L(λ₀ ^(LB))=+∞.    -   (c) Else calculate L(λ₀ ^(LB)).    -   (d) Sort users in decreasing order of w_(i)e_(i).    -   (e) Find smallest j* such that        ${\sum\limits_{i = 1}^{j^{*} - 1}N_{i}} < N \leq {\sum\limits_{i = 1}^{j^{*}}N_{i}}$    -   (f) Define n₀* $n_{0,i}^{*} = \left\{ \begin{matrix}        N_{i} & {i < j^{*}} \\        {N - {\sum\limits_{i = 1}^{j - 1}N_{i}}} & {i = j^{*}} \\        0 & {i > j^{*}}        \end{matrix} \right.$    -   (g)In case of a tie for j pick the user with the larger w_(i)        and if the w_(i)s are also equal, pick either.    -   (h) If n_(j,0)=N_(j)* then        $P_{req} = {\sum\limits_{i = 1}^{j^{*}}{\frac{n_{i,0}}{e_{i}}{\min\left( {\left( {\frac{w_{i}e_{i}}{w_{j^{*} + 1}e_{j^{*} + 1}} - 1} \right),s_{i}} \right)}}}$    -   (i) Else        $P_{req} = {\sum\limits_{i = 1}^{j^{*}}{\frac{n_{i,0}}{e_{i}}{\min\left( {\left( {\frac{w_{i}e_{i}}{w_{j^{*}}e_{j^{*}}} - 1} \right),s_{i}} \right)}}}$    -   (j) If P_(req)≧P, then go to Step 2.    -   (k) If n_(j,0)*=N_(j)* then λ₀ ^(UB)=w_(j+1)*e_(j+1)*, else λ₀        ^(UB)=w_(j)*e_(j)*.    -   (l) Calculate L₀ ^(UB)=L(λ₀ ^(UB)).    -   (m) If L₀ ^(LB)≦L₀ ^(UB), then λ₀*=λ₀ ^(LB); L₀*=L₀ ^(B).    -   (n) Else λ₀*=λ₀ ^(UB); L₀*=L₀ ^(UB).

Step 1: Iterate For ( k=0, k<iteration limit, k++)

-   -   (a) If L_(k)*>min(L_(k) ^(LB),L_(k) ^(UB)), then error!    -   (b) Calculate λ_(k+1) ^(CAND)=λ*(n_(k)*))    -   (c) Calculate L_(k+1) ^(CAND)=L(λ_(k+1) ^(CAND)), n_(k+1)        ^(CAND)=n*(λ_(k+1) ^(CAND)).    -   (d) Calculate λ*(n_(k+1) ^(CAND)), p_(k+1) ^(CAND)=p*(n_(k+1)        ^(CAND))    -   (e) Calculate V(n_(k+1) ^(CAND)).    -   (f) If V(n_(k+1) ^(CAND))>(1−[epsilon])min{L_(k+1)        ^(CAND),L_(k)*)}, then go to Step 2.    -   (g) If L_(k+1) ^(CAND)<L_(k)* and λ_(k+1) ^(CAND)>λ_(k)        ^(LB)+[epsilon](λ_(k) ^(UB)−λ_(k) ^(LB)) and λ_(k+1)        ^(CAND)<λ_(k) ^(UB)−[epsilon](λ_(k) ^(UB)−λ_(k) ^(LB)) and        λ_(k+1) ^(CAND)≠λ_(k)*    -   TRY FASTER TRACK        -   i. If f λ_(k+1) ^(CAND)<λ_(k)*            λ_(k+1) ^(UB)=λ_(k)*            λ_(k+1)*=λ_(k+1) ^(CAND)            λ_(k+1) ^(LB)λ_(k) ^(LB)        -   ii. Else            λ_(k+1) ^(LB)=λ_(k)*            λ_(k+1)*=λ_(k+1) ^(CAND)            λ_(k+1) ^(UB)=λ_(k) ^(UB)    -   (h) Else    -   DO BISECTION        -   i. If λ_(k)*−λ_(k) ^(LB)≧λ_(k) ^(UB)−λ_(k)* then            -   A. Assign                λ_(k+1) ^(CAND)=(λ_(k) ^(LB)+λ_(k)*)/2            -   B. Calculate                L _(k+1) ^(CAND) =L(λ _(k+1) ^(CAND)), n _(k+1) ^(CAND)                =n*(λ _(k+1) ^(CAND)).            -   C. Calculate                λ*(n _(k+1) ^(CAND)), p _(k+1) ^(CAND) =p*(n _(k+1)                ^(CAND))            -   D. Calculate                V(n_(k+1) ^(CAND)).            -   E. If V(n_(k+1) ^(CAND))>(1−[epsilon])min{L_(k+1)                ^(CAND, L) _(k)*)}, then go to Step 2.            -   F. If L_(k+1) ^(CAND)≦L_(k)*, then change the algorithm                state as follows                λ_(k+1) ^(UB)=λ_(k)*                L _(k+1) ^(U) B=L _(k)*                λ_(k+1)*=λ_(k+1) ^(CAND)                L _(k+1) *=L _(k+1) ^(CAND)                n _(k+1) *=n _(k+1) ^(CAND)                λ_(k+1) ^(LB)=λ_(k) ^(LB)                L _(k+1) ^(LB) =L _(k) ^(LB)            -   G. Else change as follows                λ_(k+1) ^(LB)=λ_(k+1) ^(CAND)                L _(k+1) ^(L) B=L _(k+1) ^(CAND)                λ_(k+1)*=λ_(k)*                L _(k+1) *=L _(k)*                n _(k+1) *=n _(k)*                λ_(k+1) ^(UB)=λ_(k+1) ^(UB)                L _(k+1) ^(UB) L _(k) ^(UB)        -   ii. Else            -   A. Assign                λ_(k+1) ^(CAND)=(λ_(k) ^(UB)+λ_(k)*)/2            -   B. Calculate                L _(k+1) ^(CAND) =L(λ _(k+1) ^(CAND)), n _(k+1) ^(CAND)                =n*(λ_(k+1) ^(CAND)).            -   C. Calculate                λ*(_(k+1) ^(CAND)), p _(k+1) ^(CAND) =p*(n _(k+1)                ^(CAND))            -   D. Calculate                V(n_(k+1) ^(CAND)).            -   E. If V(n_(k+1) ^(CAND))>(1−[epsilon])min{L_(k+1)                ^(CAND),L_(k)*)}, then go to Step 2.            -   F. If L_(k+1) ^(CAND)≦L_(k)*, then change the algorithm                state as follows                λ_(k+1) ^(LB)=λ_(k)*                L _(k+1) ^(LB) =L _(k)*                λ_(k+1)*=λ_(k+1) ^(CAND)                L*=L _(k+1) ^(CAND)                n _(k+1) *=n _(k+1) ^(CAND)                λ_(k+1) ^(UB)=λ_(k) ^(UB)                L _(k+1) ^(UB) =L _(k) ^(UB)            -   G. Else change as follows                λ_(k+1) ^(LB)=λ_(k+1) ^(LB)                L _(k+1) ^(LB) =L _(k+1) ^(LB)                λ_(k+1)*=λ_(k)*                L _(k+1) *=L _(k)*                n _(k+1) *=n _(k)*                λ_(k+1) ^(UB)=λk+1 ^(CAND)                L _(k+1) ^(CAND) =L _(k+1) ^(CAND)

Step 2:

-   -   (a) Round n*—note two users may have non-integer allocations in        the case of tie. In that case round down the one with the        greater power per code requirement and round up the other one.    -   (b) Recalculate p*(n*) taking all constraints including rate        constraints into account.    -   (c) Calculate the rate r*=r(p*,n*) given by        $r_{i}^{*} = {\frac{240000}{\ln\quad 2}n_{i}^{*}\quad{\ln\left( {1 + \frac{p_{i}^{*}e_{i}}{n_{i}^{*}}} \right)}}$        and TBS=r_(i)×0.002. Round down to nearest byte or some even        coarser quantization.        We can also skip steps (b)-(g) in Step 1 to get a pure bisection        based search algorithm.

2.1.4.2 Second Embodiment for Finding λ*

The second embodiment for finding the optimal A uses subgradients ofL(λ) to aid in its minimization. For any given λ, a subgradient caneasily be found as follows:

-   -   1. First find n*(λ) in equation (18). In case of a “tie” all        feasible code allocations which satisfy the sum power constraint        with equality and optimize the Lagrangian (as described after        equation (18)) are considered.    -   2. Next, using n*(λ) find the corresponding p* as given by        equation (12)    -   3. Then $P - {\sum\limits_{i}^{\quad}\quad p_{i}^{*}}$        is a subgradient to L(λ) at λ.    -   4. Since L(λ) is convex, λ* will be optimal if and only if 0 is        a subgradient of L(λ*).    -   5. Thus, we can find the optimal λ* that minimizes L(λ) by        numerically searching for λ with a 0 subgradient.

2.2 OTHER EMBODIMENTS

Based on the above results scheduler/optimizer 101 determinesappropriate values for r, n, and p as follows:

-   -   1. Collect all the input parameters P, N, N, R, R/N, P, S, e, W,        Q, c, b. (Step 301)    -   2. Calculate n. (Step 303)    -   3. Calculate optimal p and r keeping n fixed. (Step 305)

This is illustrated in FIG. 3, where FIG. 3 is a flow chart showingoperation of scheduler/optimizer 101 in accordance with anotherembodiment of the present invention. The logic flow begins at step 301where P, N, N, R, R/N, P, S, e, W, Q, c, and b are collected by theoptimizer 101. Note that steps 2-5 in the first embodiment were reallyaids to calculating the optimal n. We consider any other method tocalculate n based on the input parameters P, N, N, R, R/N, P, S, e, W,W, Q, c, and b (Step 303). Then, r and p are calculated keeping n fixed(step 305). In particular n is fixed to the value obtained in Step 303,and p and r are calculated as per equations (21) and (2) taking theactual rate constraint R into account. Finally, at step 307 values forn, p and r are output to controller 103 and utilized to schedule andtransmit data via transmitters 105.

An alternative method of calculating n based on the input parameters e,P, N, N, and S, is presented below:

-   -   1. Sort users based on any metric that is a function of the        input parameters including w_(i) and e_(i).    -   2. Find smallest j* such that        ${\sum\limits_{i = 1}^{j^{*} - 1}\quad N_{i}} < N \leq {\sum\limits_{i = 1}^{j^{*}}\quad N_{i}}$    -   3. Define n₀ $n_{0,i}^{*} = \left\{ \begin{matrix}        N_{i} & {i < j^{*}} \\        {N - {\sum\limits_{i = 1}^{j - 1}\quad N_{i}}} & {i = j^{*}} \\        0 & {i > j^{*}}        \end{matrix} \right.$    -   4. In case of a tie for j* pick the user with the larger w_(i)        and if the w_(i)s are also equal, pick either.    -   5. Optionally, we can also run one or more iteration of the        algorithm described in the previous section.

While the invention has been particularly shown and described withreference to particular embodiments, it will be understood by thoseskilled in the art that various changes in form and details may be madetherein without departing from the spirit and scope of the invention.For example, although examples where given above with respect toimplementations within an HSDPA system, one of ordinary skill in the artwill recognize that the above embodiments may be implanted in anycommunication system where codes and/or power need to be allocated. Itis intended that such changes come within the scope of the followingclaims.

1. A method for resource allocation, the method comprising: receiving amaximum total system transmit power (P); receiving a maximum number ofsystem codes available (N); receiving a maximum number of codes for eachuser in the system (N); receiving a maximum SINR value (S) for each userin the system; receiving a SINR per watt of transmit power value foreach user in the system (e); determining an optimal number of codes peruser (n) and optimal power levels per user (p) based on P, N, N, S, ande; and scheduling users with operating parameters based on n and p. 2.The method of claim 1 further comprising determining an optimal rate foreach user (r) based on n and p.
 3. The method of claim 1 whereindetermining n and p comprises maximizing a dot product of a weightvector as a function of a throughput/queue lengthw_(t)=(w₁(W_(1,t)Q_(1,t)), . . . , w_(d) (W_(d,t)Q_(d,t))) and a ratevector r_(t) to be selected at a scheduling time t, where:w_(t)=(w_(1,t), . . . , w_(d,t)) is a weight vector, W_(t)=(W_(1,t), . .. , W_(d,t)) is a vector of estimates of user throughputs,Q_(t)=(Q_(1,t), . . . , Q_(d,t)) is the vector of user queue lengths ordelays, r_(t)=(r_(1,t), . . . , r_(d,t)) is a rate vector selected, andd is the number of users.
 4. The method of claim 1 wherein the step ofdetermining n and p comprises the step of determining:V*:=max_((n,p)εχ) V(n,p) subject to $\begin{matrix}{{\sum\limits_{i}^{\quad}\quad n_{i}} \leq N} \\{{\sum\limits_{i}^{\quad}\quad p_{i}} \leq P} \\{where} \\{{{V\left( {n,p} \right)}:={\sum\limits_{i}^{\quad}\quad{w_{i}n_{i}{\ln\left( {1 + \frac{p_{i}e_{i}}{n_{i}}} \right)}}}},} \\{\chi:=\left\{ {{{\left( {n,p} \right) \geq 0}:{n_{i} \leq N_{i}}},{p_{i} \leq {S_{i}{n_{i}/e_{i}}\quad{\forall i}}}} \right\}}\end{matrix}$ w is the vector of weights w_(i)s, e is the vector of SINRper watt of transmit power e_(i)s, S is the vector of maximum SINRS_(i)s, w_(i)=ƒ_(i)(W_(i),Q_(i)), and where W=(W₁, . . . , W_(d)) isestimated throughput vector for all the users, Q=(Q₁, . . . , Q_(d)) isqueue length or delay vector for all the users, and ƒ_(i) is somefunction of the throughput and/or queue length for user i and it mayalso depend on the QoS class and parameters for user i.
 5. A method forresource allocation, the method comprising: determining a value for anoptimal number of codes per user (n) within a communication system; anddetermining an optimal rate per user (r) and an optimal power per user(p) based on n.
 6. The method of claim 5 wherein determining the numberof codes per user (n) comprises: calculating n=n*(λ*,μ*(λ*)); where λ*is an optimal Lagrange multiplier for a power constraint; μ* (λ) is aoptimal Lagrange multiplier μ for a code constraint (as a function ofthe Langrange multiplier for the power constraint λ), and n* (λ, μ) isan optimal code allocation as a function of λ,μ.
 7. The method of claim5 wherein the step of determining the optimal rate per user (r) and anoptimal power per user (p) based on n comprises the step of: calculatingn=p*(n)=p*(n, λ*(n)); where λ*(n) is an optimal Lagrange multiplier forthe power constraint; and p*(n, λ) is the optimal power allocation as afunction of n, λ.
 8. The method of claim 5 wherein the step ofdetermining the value for n, comprises the step of receiving apre-determined value for n.
 9. The method of claim 5 wherein the step ofdetermining the value for n, comprises the steps of: sorting users indescending order based on a metric that is a function of w_(i) ande_(i), where e_(i) is a SINR per watt of transmit power value and w_(i)is the gradient/weight for each user i in the system; determining asmallest j* such that${\sum\limits_{i = 1}^{j^{*} - 1}\quad N_{i}} < N \leq {\sum\limits_{i = 1}^{j^{*}}\quad N_{i}}$determining n such that $n_{i} = \left\{ {\begin{matrix}N_{i} & {i < j^{*}} \\{N - {\sum\limits_{i = 1}^{j - 1}\quad N_{i}}} & {i = j^{*}} \\0 & {i > j^{*}}\end{matrix}.} \right.$
 10. An apparatus comprising: a schedulerreceiving a maximum total system transmit power (P), a maximum number ofsystem codes available (N), a maximum number of codes for each user inthe system (N), a maximum SINR value (S) for each user in the system,and a SINR per watt of transmit power value for each user in the system(e), the scheduler outputting an optimal number of codes per user (n)and optimal power levels per user (p) based on P, N, N, S, and e. 11.The apparatus of claim 10 further comprising: a controller schedulingtransmissions to users with physical layer operating points based on nand p.
 12. The apparatus of claim 10 wherein n and p are determined bymaximizing a dot product of a weight vector as a function of thethroughput/queue length w_(t)=(w₁(W_(1,t)Q_(1,t)), . . . ,w_(d)(W_(d,t)Q_(d,t))) and a rate vector r_(t) to be selected at ascheduling time t, where: w_(t)=(w_(1,t), . . . , w_(d,t)) is a weightvector, W_(t)=(W_(1,t), . . . , W_(d,t)) is a vector of estimates ofuser throughputs, Q_(t)=(Q_(1,t), . . . , Q_(d,t)) is the vector of userqueue lengths, r_(t)=(r_(1,t), . . . , r_(d,t)) is a rate vectorselected, and d is the number of users.
 13. The apparatus of claim 10wherein n and p determined by computingV*:=max_((n,p)εχ) V(n,p) subject to $\begin{matrix}{{\sum\limits_{i}^{\quad}\quad n_{i}} \leq N} \\{{\sum\limits_{i}^{\quad}\quad p_{i}} \leq P} \\{where} \\{{{V\left( {n,p} \right)}:={\sum\limits_{i}^{\quad}\quad{w_{i}n_{i}{\ln\left( {1 + \frac{p_{i}e_{i}}{n_{i}}} \right)}}}},} \\{\chi:=\left\{ {{{\left( {n,p} \right) \geq 0}:{n_{i} \leq N_{i}}},{p_{i} \leq {S_{i}{n_{i}/e_{i}}\quad{\forall i}}}} \right\}}\end{matrix}$ e is the vector of SINR per watt of transmit power e_(i)s,w_(i)=ƒ_(i)(W_(i),Q_(i)), and where W=(W₁, . . . , W_(d)) is estimatedthroughput vector for all the users, Q=(Q₁, . . . , Q_(d)) is queuelength vector for all the users, and ƒ_(i) is a function of a throughputand/or queue length for user i.